In a previous post I mumbled a bit about perfectoid covering spaces of rigid analytic varieties. In this post, I want to sketch a fun special case.
Fix an algebraically closed nonarchimedean field. Recall the following from Scholze-Weinstein: given a cofiltered inverse system of rigid spaces over with qcqs transition maps, if there is a perfectoid space such that , then is unique up to unique isomorphism. In this scenario, I will often write when such a perfectoid space exists.
Now, fix a smooth proper curve over of genus , with associated adic space over .
Lemma. There is a natural equivalence .
Proof: This follows from Lutkebohmert’s -adic Riemann existence theorem, or from a very cool theorem of Fresnel and Matignon: any irreducible, quasicompact, separated one-dimensional rigid analytic space is either affinoid or projective.
Now fix a base point , and let denote the usual étale fundamental group from SGA; this is a profinite group depending only on (up to isomorphism), and there is a natural fiber functor
where -Sets (for a profinite group) denotes the category of finite sets with continuous (left) -action. For any open subgroup , we have a corresponding connected finite étale cover (which is uniquely algebraizable to a covering ).
Theorem. Suppose the Néron model of is an abelian scheme over . Then there is a natural perfectoid space such that
Under the natural map , is naturally a -torsor over , and any can be recovered from as the categorical quotient .
In the remainder of this post, I’ll sketch the construction of . Here’s the idea: Let be the rigid analytic Jacobian of , so our point gives a natural closed immersion . On fundamental groups we get a natural surjection
Set , and let denote the associated covering of ; these form an inverse system in the obvious way. We are going to first construct a perfectoid space such that , and then carefully pile all the remaining ‘s on top of , using the almost purity theorem to ensure that this pileup remains perfectoid.
Let denote a copy of mapping to under , so is a finite étale covering of degree . These form an inverse system in the obvious way.
Lemma. The map is obtained from the embedding by pulling back under .
Proof: Easy exercise.
Lemma. There is a perfectoid space such that .
Proof: This is Lemme A.16 in Pilloni-Stroh’s “Cohomologie coherent et representations Galoisiennes”.
Lemma (“The pullback lemma”). Let be a morphism of rigid analytic spaces, and let be a perfectoid space. If is “blah”, where “blah” open immersion, closed immersion, finite étale, étale, then exists as a perfectoid space and is “blah”.
If for some cofiltered inverse system of rigid spaces with qcqs transition maps, then .
Proof: It suffices to work locally on and . We give a brief sketch:
Open immersion: Immediate.
Closed immersion: Pull back the ideal sheaf defining in and argue as in § II.2 of Scholze’s torsion paper.
Finite étale: Immediate from the almost purity theorem.
Etale: Any such morphism is locally a composite of finite étale morphisms and open immersions, so we reduce to those cases.
Lemma. The fiber product exists as a perfectoid space, and .
Proof: Immediate from the pullback lemma applied to the case where , , .
Lemma. For any open subgroup , there is a natural perfectoid space such that , and is finite étale over .
Proof: The space is naturally a connected component of ; by the pullback lemma, this fiber product is perfectoid and finite étale over . The inclusion map is finite étale, so we deduce that is finite étale.
The spaces form a cofiltered inverse system of perfectoid spaces over , with finite étale transition maps.
Lemma. The inverse limit
exists as a perfectoid space.
Proof: Argue locally on : let be an open affinoid perfectoid subset, with preimage . By the definition of finite étale morphisms, each is affinoid perfectoid; since the category of affinoid perfectoid spaces admits all limits, we get as an affinoid perfectoid space over .
Final lemma. Let be the perfectoid space constructed in the previous lemma. Then .
Proof: Easy consequence of all the above.